Solving Rota's Conjecture

Abstract

I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution to Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of linear independence among vectors; given a finite collection of vectors in a vector space, each subset is either dependent or independent. A matroid consists of a finite ground set together with a collection of subsets that we call independent; the independent sets satisfy natural combinatorial axioms coming from linear algebra. Not all matroids can be represented by a collection of vectors and, ever since their introduction by Hassler Whitney [26] in 1935, mathematicians have sought ways to characterize those matroids that are. Rota’s Conjecture asserts that representability over any given finite field is characterized by a finite list of obstructions. We will formalize these notions, and the conjecture, in the next section. In the remainder of this introduction, we will describe the journey that led us to a solution. In the late 1990s, Rota’s Conjecture was already known to hold for fields of size two, three, and